The article deals with normal linear operators B with a unit spectral radius in Hilbert spaces, for which the successive approximations x_n+1 = Bx_n + f with an arbitrarily initial approximation x₀ converge to a solution of the equation x = Bx + f (under condition that these solutions exist). Sufficient conditions for the convergence of successive approximations on subspaces of source-wise represented functions and in weakened norms are established. The behavior of residuals and corrections of these approximations is studied, too. Moreover, the behavior of “approximate” successive approximations is also investigated.

In this article we consider the distribution of algebraic numbers in very short intervals. The concept of a type of a short intervals is introduced. Depending on the type of the interval the estimations for the number of algebraic numbers within the interval are found.

Analogs of interpolation splines for functions of matrix variable, defined on sets of matrices with the ordinary, Jordan, Hadamard and Frobenius multiplication, has been constructed. Some of the obtained formulas contain Gateaux differentials of the interpolated function.

A method for obtaining tiles of operations of a parallel algorithm is developed. Propositions for ranking tiles size parameters are stated and proved. Statements to assess the amount of communication operations generated by the partition of the set of iterations are stated and proved.

In the present article we consider the interaction between the solution of the problem of obtaining necessary and sufficient conditions of existence of a time-independent first integral analytical in the neighborhood of the origin for the complex system of the form x = y + P(x, y), y = −x + Q(x, y), where P(x, y), Q(x, y) are polynomials without constant and linear terms, and the solution of the center-focus problem for the real system of the same form.

In present work the questions related to decreasing the control voltage of amplitude Fabry–Perot modulators with conserving their high efficiency have been considered. Schematic models of high-efficient amplitude Fabry–Perot electrooptical modulators have been proposed.

Spectral and kinetic characteristics of phosphorescence and fluorescence of the indole solution in ethanol at 77 K have been studied. The decrease of phosphorescence quantum yield at red – edge excitation has been observed.

The magnetic and magnetotransport properties of the La_0.45Ba_0.55CoO_3–x cobaltites have been studied as a function of oxygen content. It is shown that the decrease in the oxygen content leads to strong enhancement of the magnetoresistance and the metamagnetic behavior. It is suggested that the colossal magnetoresistance results from the spin state transition induced by an external magnetic field.

Deep level transient spectroscopy (DLTS) has been used for studying the electrically active defects in p-type silicon crystals irradiated with fast electrons and α-particles. A new bistable radiation-induced center with deep levels around the midgap of silicon is revealed. The main characteristics of this center are determined and some suggestions on its origin are given.

A new approach to the synthesis of pyrazole-containing liquid crystals was investigated. This synthetic approach is based on the mesogenic reaction of 3-aryl-3-ketoesters with 4-substituted arylhydrazines. As an example of this approach, the reaction of 3-aryl-3-ketoesters with 4-methoxyphenylhydrozine was investigated. As a result, a series of new mesogenic pyrazoles were synthesized. Substances with a monotropic nematic phase were detected among the synthesized compounds.

The possibility of application of thin films of molybdenum-vanadium oxide as the photoresist material was demonstrated. The photosensitivity of such a system was due to the photo-induced polycondensation of polymolybdic acid oligomers in xerogel V₂O₅ : MoO₃ films.

FTIR spectroscopy analysis of the organic reagent content (a mixture of amine to extract oil) in samples of potassium chloride was done and the main metrological characteristics were determined. To improve the accuracy of the analysis of the spectral data on the processing and construction of calibration curves, it is proposed to use second-degree non-linear polynomial equations and to take into account the optical density of several absorption bands.

For the first time the linear sizes of a rare and protected species Eurytemora laсustris have been obtained for all ages. The range of body sizes of adult animals from the lakes of Belarus is within the size limits indicated in the literature on the water bodies of Russia and the countries of Western Europe. It has been found that the body length of adult males exceeds that of females, which is not typical for other species of planktonic freshwater copepods and contradicts the literature data. The established fact is a distinctive feature of this species.

Naturalization of a monkey goby Neogobius fluviatilis (Pallas, 1814) for the basin of the Neman River has been shown. The new way of invasion of the Ponto-Caspian fishes to the basin of the Baltic Sea has been revealed. It has been established that the alien species N. fluviatilis is a dominant fish species in littoral zone of Vileysky reservoir.

The nature of the interaction between structural and functional parameters of a leaf, which discloses the mechanism of adaptation of tropical and subtropical woody species in greenhouse conditions, is revealed.

The effect of exogenous 5-aminolevulinic acid (ALA) on the growth and activity of nitrate reductase (NR, EC 1.6.6.1), the synthesis of proline and the superoxide anion radical generation in barley (Hordeum vulgare L. cv. Gonar) seedlings grown under salinization with NaCl has been studied. The obtained data indicated that ALA (20, 40 and 80 mg L^–1) increased the elongation and leaf blade expansion of barley seedlings exposed to 150 mM NaCl as compared to control plants, which were not treated with ALA. In the presence of the NR inductor – its substrate 20mM KNO₃, exogenous ALA stimulated NR gene Nar 1 expression, increased the NR-protein content and enzymatic activities in plants grown in 150 mM NaCl solution. ALA also induced the accumulation of free proline and decreased the generation of superoxide anion radicals. Thus, this study demonstrates that ALA effects on salt-stressed barley seedlings involves the stimulation of NR activities both at transcriptional and translational levels and leads to complex physiological modifications, such as the improvements of plant growth and antiradical defense.

Changes in the surrounding tissues in the late postoperative period associated with the implantation of composite mesh prosthesis (Parietene ProGrip) consists in a combination of acute reactions for the introduction of a foreign body and the manifestations of chronic granulomatous inflammation, which means the isolation of the material by a connective tissue capsule, failure and its removal outside the body. In the course of time, the severity of these reactions is reduced, but not completely leveled.

Five types of territories differing in the values of the volume activity of the «local» (earth) radon and in the degree of their radon hazard have been distinguished within the soil air region.

Oxide-ceramic coatings during the process of Micro-Arc Oxidation (MAO) with nanoscale titanium nitride were modified, which allowed one to obtain oxide-nitride ceramics in the shape of a coating on an aluminum alloy D16. The highest concentration of titanium nitride was registered in the surface coating layers, and the modification area extended to 1/3 of its thickness. It was found that the modification was accompanied by a 1.2–1.4-fold increase in the microhardness that achieved 23 GPa. Tribological tests under boundary lubrication showed that the modification of the MAO-coating with titanium nitride reduced the friction coefficient by a factor of 1.5–2.8 as compared to an unmodified coating. In addition to improving anti-friction properties, the wear resistance of coatings increased by a factor of 3–5.

The article analyzes the "Autobiography" by Bоgyslaw Radzivill that is one of the most outstanding literary monument of the XVII century history of the Polish-Lithuanian Commonwealth. The duke’s memoirs were studied as an authentic source on the history of Polish Crown and Grand Duchy of Lithuania and as the duke’s walk of life. The analysis made gives a fair opportunity to specify the role of B. Radzivill in the political history of the country.

The article proves that the industrial-type mushroom production remains an underestimated branch of the Belarusian agriculture, as well as the economic and social usefulness of its development that is determined by a number of factors: the availability of appropriate natural and economic environment; a high export potential due to its geographic and product diversification; a potential for import substitution and domestic market expansion; a relatively high effectiveness of production and investment use; social importance of consuming cultivated mushrooms as wholesome ecologically pure foodstuffs; a potential for job creation in the rural area.